Analysis of Rational Function: The function f(x) = (10x – 30)/(x² – x – 6) is studied for its characteristics:
- Factorization: f(x) = 10(x – 3)/((x – 3)(x + 2))
- Vertical Asymptotes (VA): At x = -2.
- Holes: A hole is present at (3, 2).
- Horizontal Asymptotes (HA): Asymptote lies at y = 0.
- X-Intercepts: No x-intercept due to a hole at x = 3.
- Domain: (-∞, -2) ∪ (-2, 3) ∪ (3, ∞).
- Range: (-∞, 0) ∪ (0, 2) ∪ (2, ∞).
- Y-Intercept: At (0, 5).
- Additional Points: The point (-3, -10) is on the graph, not crossing the x-axis.
This structured approach aids in graphing rational functions and understanding their graphical properties.
Graphing A Rational Function With A Vertical Asymptote And Holes